ZETA FUNCTION REGULARIZATION OF PATH INTEGRALS IN CURVED SPACETIME
ZETA FUNCTION REGULARIZATION OF PATH INTEGRALS IN CURVED SPACETIME
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DOI：
10.1007/bf01626516
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发表时间：
1977-01-01
影响因子：
2.4
通讯作者：
HAWKING, SW
中科院分区：
文献类型：
--
作者：
HAWKING, SW
本文描述了一种在弯曲背景时空上正则化二次路径积分的方法。从作用积分中出现的微分算子的特征值形成一个广义的ζ函数。函数是一个亚纯函数，它在原点的梯度被定义为算子的行列式。这种技术与维度正则化一致，即通过添加额外的平面维度来推广维度。广义zeta函数可以表示为描述在参数时间的第五维四维时空流形上扩散的热方程核的Mellin变换。利用热核的渐近展开，可以推导出路径积分在背景度规的尺度变换下的行为。这表明在所有黑洞背景度量的积分中可能存在一个自然的截止点。通过对路径积分进行函数微分，可以得到一个即使在黑洞视界上也是有限的能量动量张量。这个能量动量张量有一个反常的轨迹。
This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises tondimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.